Theorem cfinufil 23884

Description: An ultrafilter is free iff it contains the Fréchet filter cfinfil 23849 as a subset. (Contributed by NM, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
cfinufil	⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ Fin} ⊆ 𝐹))

Distinct variable groups:   𝑥,𝐹   𝑥,𝑋

Proof of Theorem cfinufil

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpwi 4563	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
2		ufilb 23862	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (¬ 𝑥 ∈ 𝐹 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹))
3	2	adantr 480	. . . . . . . . 9 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑋 ∖ 𝑥) ∈ Fin) → (¬ 𝑥 ∈ 𝐹 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹))
4		ufilfil 23860	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
5	4	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → 𝐹 ∈ (Fil‘𝑋))
6		filfinnfr 23833	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑋 ∖ 𝑥) ∈ 𝐹 ∧ (𝑋 ∖ 𝑥) ∈ Fin) → ∩ 𝐹 ≠ ∅)
7	6	3exp 1120	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ((𝑋 ∖ 𝑥) ∈ Fin → ∩ 𝐹 ≠ ∅)))
8	7	com23 86	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑋 ∖ 𝑥) ∈ Fin → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ∩ 𝐹 ≠ ∅)))
9	5, 8	syl 17	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ Fin → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ∩ 𝐹 ≠ ∅)))
10	9	imp 406	. . . . . . . . 9 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑋 ∖ 𝑥) ∈ Fin) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ∩ 𝐹 ≠ ∅))
11	3, 10	sylbid 240	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑋 ∖ 𝑥) ∈ Fin) → (¬ 𝑥 ∈ 𝐹 → ∩ 𝐹 ≠ ∅))
12	11	necon4bd 2953	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑋 ∖ 𝑥) ∈ Fin) → (∩ 𝐹 = ∅ → 𝑥 ∈ 𝐹))
13	12	ex 412	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ Fin → (∩ 𝐹 = ∅ → 𝑥 ∈ 𝐹)))
14	13	com23 86	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (∩ 𝐹 = ∅ → ((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)))
15	1, 14	sylan2 594	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → (∩ 𝐹 = ∅ → ((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)))
16	15	ralrimdva 3138	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ → ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)))
17	4	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
18		uffixsn 23881	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → {𝑦} ∈ 𝐹)
19		filelss 23808	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ {𝑦} ∈ 𝐹) → {𝑦} ⊆ 𝑋)
20	17, 18, 19	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → {𝑦} ⊆ 𝑋)
21		dfss4 4223	. . . . . . . . . . 11 ⊢ ({𝑦} ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ {𝑦})) = {𝑦})
22	20, 21	sylib 218	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (𝑋 ∖ (𝑋 ∖ {𝑦})) = {𝑦})
23		snfi 8992	. . . . . . . . . 10 ⊢ {𝑦} ∈ Fin
24	22, 23	eqeltrdi 2845	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin)
25		difss 4090	. . . . . . . . . . 11 ⊢ (𝑋 ∖ {𝑦}) ⊆ 𝑋
26		filtop 23811	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
27		elpw2g 5280	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐹 → ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ {𝑦}) ⊆ 𝑋))
28	17, 26, 27	3syl 18	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ {𝑦}) ⊆ 𝑋))
29	25, 28	mpbiri 258	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋)
30		difeq2 4074	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑋 ∖ {𝑦}) → (𝑋 ∖ 𝑥) = (𝑋 ∖ (𝑋 ∖ {𝑦})))
31	30	eleq1d 2822	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑋 ∖ {𝑦}) → ((𝑋 ∖ 𝑥) ∈ Fin ↔ (𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin))
32		eleq1 2825	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑋 ∖ {𝑦}) → (𝑥 ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
33	31, 32	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑋 ∖ {𝑦}) → (((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) ↔ ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
34	33	rspcv 3574	. . . . . . . . . 10 ⊢ ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
35	29, 34	syl 17	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
36	24, 35	mpid 44	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → (𝑋 ∖ {𝑦}) ∈ 𝐹))
37		ufilb 23862	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ {𝑦} ⊆ 𝑋) → (¬ {𝑦} ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
38	20, 37	syldan 592	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (¬ {𝑦} ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
39	18	pm2.24d 151	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (¬ {𝑦} ∈ 𝐹 → ¬ 𝑦 ∈ ∩ 𝐹))
40	38, 39	sylbird 260	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → ((𝑋 ∖ {𝑦}) ∈ 𝐹 → ¬ 𝑦 ∈ ∩ 𝐹))
41	36, 40	syld 47	. . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → ¬ 𝑦 ∈ ∩ 𝐹))
42	41	impancom 451	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)) → (𝑦 ∈ ∩ 𝐹 → ¬ 𝑦 ∈ ∩ 𝐹))
43	42	pm2.01d 190	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)) → ¬ 𝑦 ∈ ∩ 𝐹)
44	43	eq0rdv 4361	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)) → ∩ 𝐹 = ∅)
45	44	ex 412	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → ∩ 𝐹 = ∅))
46	16, 45	impbid 212	. 2 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ ↔ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)))
47		rabss 4024	. 2 ⊢ ({𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ Fin} ⊆ 𝐹 ↔ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹))
48	46, 47	bitr4di 289	1 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ Fin} ⊆ 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  {crab 3401   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  {csn 4582  ∩ cint 4904  ‘cfv 6500  Fincfn 8895  Filcfil 23801  UFilcufil 23855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1o 8407  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fbas 21318  df-fg 21319  df-fil 23802  df-ufil 23857

This theorem is referenced by: (None)